Electrostatic voltmeters are often constructed as shown in Fig. 3P.13a. N pairs of pie-shaped plates form the stator and rotor of a variable capacitor (Fig. 3P.13a shows six pairs of rotor plates and six pairs of stator plates). The rotor plates are attached to a con- ducting shaft that is free to rotate through an angle 0. In the electrostatic voltmeter a pointer is attached to this shaft so that the deflection is indicated on a calibrated scale (not shown). (a) Determine q(v, 0), where q is the charge on the stator and is the voltage applied between the rotor and stator. The device is constructed so that fringing fields can be ignored and the area of the plates is large compared with the cross section of the shaft. In addition, it is operated in a region of 8 in which the plates overlap but not completely. (b) Find the torque of electrical origin on the rotor. (c) The shaft is attached to a torsional spring which has the deflection when Problems Vo(t) N stator plates R N rotor plates Hom Fig. 3P.13a subjected to a torque Tm, where 8 and Tm are related by TK(0-a). The shaft has a moment of inertia J and is subject to a damping torque B doldt. Write the torque equation for the shaft. (d) The circuit of Fig. 3P.136 is attached to the terminals. Write the electrical equation for the system. [The results of parts (c) and (d) should constitute two equations in two unknowns.] Torsion spring Fig. 3P.136 "zero adjust" knob on the electrostatic voltmeter is used to set a in such a way that a pointer attached to the shaft indicates 0 when 0 = . A constant voltage v = V is attached to the terminals. What is the static deflection of the pointer (0-x) as a function of Vo?